# THE SHORTEST SHORT-CUT A System of Mental Arithmetic

by Oluf Nielsen

The purpose of this pamphlet is to promote an alert mind. As training for the mind, it is helpful in all studies, not only mathematics.

It does not interfere with the current prescribed courses of study and is no criticism of present instruction. It is a helpful addition to a student's mathematical ability.

If a student will memorize the squares of the numbers between 10 and 20 (only nine!), and read this pamphlet, it will give him the power to multiply large numbers such as 175 x 175.

 175 x 175 = (170 x 180) + (5 x 5) = 30,625 275 x 275 = (270 x 280) + (5 x 5) = 75,625

The reason for 280 instead of 270 is due to changing the problem to an equation such as:

 75² = (70 x 70) + (5 x 70) + (5 x 70) + (5 x 5) = 4900 + 350 + 350 + 25 = 5625

By adding 10 to the second 70, the two center sections of 350 are eliminated and it becomes:

 75² = (70 x 80) + (5 x 5) = 5625

All numbers ending in 5 can be squared in this manner.

 65² = (60 x 70) + (5 x 5) = 4225

It is fast and easy!

The student will need to memorize the squares of the numbers from 10 to 20.

10² = 10 x 10 = 100
11² = 11 x 11 = 121
12² = 12 x 12 = 144
13² = 13 x 13 = 169
14² = 14 x 14 = 196
15² = 15 x 15 = 225
16² = 16 x 16 = 256
17² = 17 x 17 = 289
18² = 18 x 18 = 324
19² = 19 x 19 = 361
20² = 20 x 20 = 400

## How to Find the Square of All Numbers to 100

Examples:

 82² = 80 x 80 + (80 + 82) (2) = 6400 + 324 = 6724 81² = 80 x 80 + (80 + 81) (1) = 6400 + 161 = 6561 Center 80² = 6400 79² = 80 x 80 - (80 + 79) (1) = 6400 - 159 = 6241 78² = 80 x 80 - (80 + 78) (2) = 6400 - 316 = 6084

Continuing this process:

 77² = 75 x 75 + (75 + 77) (2) = 5625 + 304 = 5929 76² = 75 x 75 + (75 + 76) (1) = 5625 + 151 = 5776 Center 75² = 5625 74² = 75 x 75 - (75 + 74) (1) = 5625 - 149 = 5476 73² = 75 x 75 - (75 + 73) (2) = 5625 - 296 = 5329

## How to Multiply Any Two Numbers in Any Ten Group

For example (60 to 70):

If the units added = 10:

 61 x 69 = (6 x 7) (100) + (1 x 9) = 4200 + 9 = 4209 62 x 68 = (6 x 7) (100) + (2 x 8) = 4200 + 16 = 4216 63 x 67 = (6 x 7) (100) + (3 x 7) = 4200 + 21 = 4221 64 x 66 = (6 x 7) (100) + (4 x 6) = 4200 + 24 = 4224

If the units added do not = 10, adjust to use units which = 10 and add or subtract the difference.

 63 x 69 = 61 x 69 + (2) (69) = 4209 + 138 = 4347 71 x 77 = 73 x 77 - (2) (77) = 5621 - 154 = 5467

## Multiplying by Comparative Measure Called the Use of the Bar

The bar of any number is the nearest ten to it. For example, the bar of 38 is 40 and the bar of 78 is 80.

If both numbers are under their bars, subtract a number which is equal to multiplier plus bar of multiplicand times number under the bar from product of the two bars.

For example, 78 is 2 under its bar of 80.

 78 (multiplicand) x 38 (multiplier) 78 x 38 = 40 x 80 - (80 + 38) (2) = 3200 - 236 = 2964

If both numbers are over their bars, add a number which is equal to multiplier plus bar of multiplicand times number over the bar to product of the two bars.

 82 x 42 = 80 x 40 + (42 + 80) (2) = 3200 + 244 = 3444 79 x 39 = 80 x 40 - (39 + 80) (1) = 3200 - 119 = 3081 77 x 37 = 80 x 40 - (37 + 80) (3) = 3200 - 351 = 2849

If the numbers are both over or under the bar but not in equal amounts, adjust to make them equal and add or subtract the difference.

 78 x 37 = 77 x 37 + 37 = 2849 + 37 = 2886

If the multiplicand is under the bar and the multiplier is over the bar, then add the difference between the multiplier and the bar of the multiplicand.

 78 x 42 = 80 x 40 + (80 - 42) (2) = 3200 + 76 = 3276

If the multiplicand is over the bar and the multiplier is under the bar, subtract the difference between the multiplier and the bar of the multiplicand.

 82 x 38 = 80 x 40 - (80 - 38) (2) = 3200 - 84 = 3116

A split bar is when one number is over the bar and the other is under.

Examples:

 88 x 62 = 90 x 60 + (90 - 62) (2) = 5400 + 56 = 5456 Split Bar 92 x 58 = 90 x 60 - (90 - 58) (2) = 5400 - 64 = 5336 Split Bar 148 x 92 = 150 x 90 + (150 - 92) (2) = 13500 + 116 = 13616 Split Bar 148 x 88 = 150 x 90 - (150 + 88) (2) = 13500 - 476 = 13024 Both Under 152 x 92 = 150 x 90 + (150 + 92) (2) = 13500 + 484 = 13984 Both Over

Note: Always select the bar closest to the numbers to be multiplied.

## How to Multiply Any Number by 33 1/3 or 25

To multiply by 33 1/3, as a short cut, multiply by 100 and divide by 3.

33 1/3 x 84 = 8400 / 3 = 2800

33 1/3 x 87 = 8700 / 3 = 2900

But if you multiply by 33 only, you subtract 1/100 of the answer or 28 or 29 respectively.

 33 x 84 = 8400 / 3 - 28 = 2800 - 28 = 2772 33 x 87 = 8700 / 3 - 29 = 2900 - 29 = 2871 33 x 86 = 8600 / 3 - 28 = 2866 - 28 = 2838

Note: Here remainders of 2 from 8600 / 3 were lost and were not needed.

To multiply by 25, multiply by 100 and divide by 4.

25 x 54 = 5400 / 4 = 1350

## Mulitiplication by Formula

In high school algebra:

(4x) (2x) = 8x²

In college algebra:

The answer equals half the sum squared minus half the difference squared.

 (4x) (2x) = (6x / 2)² - (2x / 2)² = 9x² - x² = 8x² 93 x 57 = (150 / 2)² - (36 / 2)² = 75² - 18² = 5625 - 324 = 5301 766 x 534 = (1300 / 2)² - (232 / 2)² = 650² - 116² = 422,500 - 13,456 = 409,044 75,500² = (75 x 76 x 1,000,000) + (5 x 5 x 10,000) = 5,700,000,000 + 250,000 = 5,700,250,000 75,500 x 80,500 = (156,000 / 2)² - (5,000 / 2)² = 78,0000² - 2,500² = 6,084,000,0000 - 6,250,000 = 6,077,750,000

When the student becomes inquisitive enough he will wonder what put the planets like our earth in orbit at 93,000,000 miles from the sun. What keeps it from being drawn into the sun?

If the north pole was not pointing towards the north star, would we have the four seasons?

At what speed do we travel to make the orbit in 365 1/4 days?

What would happen if the earth did not roll over 1,000 miles per hour establishing day and night divided in the 24 hours?

Would it work if it rolled at 500 miles per hour? Or would the nights be too cold and the days too hot for vegetation?

Wishing to know is the start of thinking. Only the drive to know will start the process. Out of more than 6,000,000 brain cells, some must be inactive. Let us call it cold storage.

Maybe with this mental arithmetic, we can get some of those cells out of cold storage!

Oluf Nielsen, Author (1891–1967)
Council Bluffs, Iowa
Circa 1965
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